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18G7.] On the Numerical Value of Euler's Constant. 429 

perty of the angle of convergence, by which the most minute differences 
in the distances of objects and the slightest relief on their surfaces can be 
detected, and by which also in the abnormal conversion introduced in its 
action by the pseudo^cope all our sensations are reversed. Therefore the 
pseudoscope is the great test of the phenomena of binocular vision ; for 
by reversing certain sensations which by constant habit we may hardly 
notice, it renders them more conspicuous by the comparison of the abnor- 
mal state brought out by its action, and proves the theory of binocular 
vision in the most effective manner. 

A truth is never better established than when it can be shown that the 
same principles are capable of producing contrary effects when they are 
applied in a contrary way. 

Professor Wheatstone, by adding the pseudoscope to the stereoscope, has 
thus in the most scientific and ingenious manner completed his splendid 
discovery, and left very little (we might almost venture to say that he has 
left nothing) for further investigations in the physiology of binocular 
vision. 



II. " On the Calculation of the Numerical Value of Euler^s Constant^ 
which Professor Price, of Oxford, calls E.^^ By William 
Shanks, Esq., Houghton-le-Spring, Durham. Communicated 
by the Rev. B. Price, P.R.S. Received March 28, 1867. 

In the year 1 853 Dr, Rutherford, of the Royal Military xicademy, Wool- 
wich, sent a paper on the Computation of the value of tt to the Royal 
Society, and the paper was published in the ' Proceedings' of that learned 
body*. The value of tt is there given to 607 decimals, the first 440 being 
the joint production of Dr. Rutherford and the author of this paper, and 
the remaining 167 decimals having been calculated by the present writer, 
for the accuracy of which he alone is responsible. Subsequently, the 
Astronomer Royal, G. B. Airy, Esq., kindly presented the author's paper 
on the Calculation of the value of e, the base* of Napier's logarithms, to 
upwards of 200 decimals ; the aforesaid paper also contained the Napierian 
logarithms of 2, 3, and 5, as well as the modulus of the common system, 
all to upwards of 200 places of decimals. This paper was not, however, 
published, but deposited in the Archives of the Royal Society ; but an abs- 
tract, containing the numerical results, was printed in the * Proceedings 'f. 
In a paper sent by the author to the Astronomer Royal, and forwarded by 
him to the Royal Society, will, the author believes, be found the reciprocal 
of the prime number 17389, consisting of a circulating period of no less 
than 1 7388 decimals, the largest on record. Some few remarks are also 
given touching circulates generally, and the easiest modes of obtaining them. 
The writer now desires to supplement what he then did, by giving the 

* Vol. vi. p. 273. t Vol. vi. p. 397. 



430 Mr. W. Shanks ow the Calculation of [April 11, 

numerical value of Euler's constant, which is largely employed in " Infi- 
nitesimal Calculus," to a greater extent than has hitherto been found, and 
free from error. 

In Crelle's Journal for 1860, vol. Ix. p. 375, M. "Oettinger has contri- 
buted an article on Euler's constant, and especially on " certain discre- 
pancies " in the value given by former mathematical writers. Adopting 
the formula there employed, as being well adapted for the purpose, the 
writer of this paper has both corrected and extended what has been pre- 
viously done ; and as very great care has been bestowed upon the calcu- 
lations, so as to exclude error, he confidently believes that his results are, 
as far as they go, absolutely correct. He may remark that, since the 
separate values of n in the formula (which, see below) produce identical 
results as far as they go, and the higher the value of n the more nearly 
we can approximate to the value of the constant, we thus have sufficient 
proof afforded of the correctness of the value found when ?^ is 10, 20, 50, 
or 100. If the writer can command sufficient leisure, he may resume the 
calculation by and by, and, making n 1000, he may thus verify, as well as 
extend, the value of Euler's constant given in this paper. The numbers 
10, 20, 50, 100, 200, and 1000, especially 10 and its integral powers, are 
more easily handled than others, particularly in those terms of the formula 
which contain Bernoulli's numbers. The harmonic progression is here 
^^ summed" much further than was requisite for finding E to 50 or 55 de- 
cimals ; but this was of some importance in ensuring correctness in the 
decimal expression of each of the higher terms of S^qq and S^qq. It may 
be observed that the numbers of decimal places in E, obtained from n being 

10, 20, 50, 100, and 200, are nearly proportional to 10% 20^, 50% lOOi 

and 200^ — a rather curious coincidence. 

The formula for Euler's constant, employed by M. Oettinger, as above 
stated, is — 

c 1 IB, B, B3 B, 

Constant = Sw—loge?i-~^4- -2^^-4^+5^6—3;^ + . • . .&c., where 

S/^= l4.|-fi-f-i-}-^4- , and Bp B^, B3, &c. are Bernoulli's numbers. 

lb 

14. 1+^4. . . . .-jV=2 -928968253 

1+4+4+.... 2V=3-59773 96571 43681 91148 37690 68908 38779 38367 

10245 37897 60291 30827 89243 77995 58542 59259 

83201 52499 71906 31865 55446 61736 61220 10084 

85838 20720 04363 02603 48526 602. 

H-i+ 2V = 3-81595 81777 53506 86913 48136 76474 73449 06181 

89635 55401 83780 86220 32609 11027 77063 20242 
32778 03544 32662 95332 52228 09675 62970 52433 
81377 45056 57669 24455 54624 85197 30818 82894 
77830 46585 03877 77968 87905 92007 71409 68243 
+ (circulating period consists of 1584 decimal places). 



1867.] the Numerical Value of Euler^s Constant, 431 

1 + J4- ^V =4-49920 53383 29425 05756 04717 92964 76909 19706 

01823 96745 38296 58902 43217 68318 06557 86735 
78157 82663 88434 12900 97472 82033 55398 82942 
03084 20093 94581 03200 90115 13091 15572 24477 
64889 95794 59834 14243 62248 23530 45299 19591 
4- (circulating period consists of 1^275,120 decimal places). 

l4-i+. -^^=5-18737 75176 39620 26080 51176 75658 25315 79089 

72126 70845 16531 76533 95658 72195 57532 56049 

66056 87768 92312 04135 49921 06986 97779 79182 

73403 18617 00828 94825 42444 49096 57618 56974 

16326 13467 07313 21114 47132 49733 09103 51089 
4- (circulating period consists of 39^419,059^680 decimal 
places). 

1+1+ 2io=5-87803 09481 21444 47605 73863 97130 86163 68374 

00246 53027 30844 64971 94472 28783 30029 84018 
15499 64301 86679 89238 37326 83211 85439 05911 
76542 77855 27568 86559 30203 06049 25715 75389 
22254 75748 47845 75246 64079 54805 61627 08837 
4- (circulating period consists of 

2,498,236,128,143,832,017,541,600 decimal places). 

The following value of Euler's constant has been found from the re- 
spective sums given above, of the 10, 20, 50, 100, and 200 first terms of 
the Harmonic Progression : — 

E or Eul. const. = -57721 56649 01532 86060 6 (last term employed is- 24^4) ' 

last term is -24^4). 

E= -57721 56649 01532 86060 65120 90082 40243 1042 (last 

term is+ ^). 

E= -57721 56649 01532 86060 65120 90082 40243 10421 

59335 9 (last term is- ,,,^,11, J. 
\ «* 24.100=^7 

E= -57721 56649 01532 86060 65120 90082 40243 10421 59335 



93995 35989 (last term is4- ^^^l 
\ 26.2002V 



Certainly 50 decimals are correct, and probably 55, in the value last 
given. 

March 2, 1867. 

Supplementary Paper to that of March 2, 1867, "On the Calculation of 
the Numerical Value of Euler^s Constant.^^ By William Shanks, 
Esq., Houghton-le-Spring, Durham. Received April 9, 1867. 

When /i=500, we have 

l + l+l 5^o=6''^9282 34299 90524 60298 92871 45367 97369 48198 

13814 39680 91166 43088 89685 43566 23790 65049 

24576 49408 73586 66039+ 



432 On the Numerical Value of Euler's Constant. [April 11, 

E = -57721 56649 01532 86060 65120 90082 40243 10421 
59335 93995 35988 05771 53865 48677 (last term 



14 



)■ 



28.500^« 

When n = 1 000, we have 

l+i + J+..T(5ViT='^'48547 08605 50344 91265 65182 04333 90017 65216 

79169 70883 36657 73626 74995 76993 49165 20244 
09599 34437 41184 50813+ 

E= -57721 56649 01532 86060 65120 90082 40243 10421 

59335 93995 35988 05772 02455 61942 00508 15825 

last term +g^)3^j. 

Hence we see that 54 decimal places are correct in the value of E (n being 
200) last given in the paper dated March 2, 1867, — also that 59 decimals 
are correct in the value of E when n=500. When ?z= 1000, probably 65 
decimals in the value of E are correct. 

When w=l, we readily find E=:*57 (last term — ^). 



j» 



=2, „ E=-57721 Mast term --^e)' 



=5, 



JJ ^> JJ 



E=:-57721 56649 015 Aast term 4-2|^^2)- 



When n= 1, E consists of 2 decimals. 

= 10, „ 21 decimals. 

= 100, „ 46 decimals. 

= 1000, „ 65 decimals, probably, 

n= 2, „ 5 decimals. 

20, ^, 28 decimals. 

200, „ 54 decimals. 

„ n= 5, „ 13 decimals. 

JJ 50, „ 39 decimals. 

500, „ 59 decimals. 



JJ 
>j 
>j 
JJ 
JJ 
j» 



JJ 



From the above we may fairly infer that when n is increased in a geome- 
trical ratio, the corresponding number of decimals obtained in the value 
of E increases only in something like an arithmetical one, and that pro< 
bably from 50,000 to 100,000 terms in the Harmonic Progression would 
require to be summed in order to obtain 100 places of decimals in the value 
of E, Euler's constant.